#
# Numerical dsolve is used with Euler's method.  The numeric solutions  
# are graphed and compared with the analytic solution.
#
# Maple Worksheet written by Denise Kirschner
# 
> with(DEtools): with(plots):  
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# 
> p1:=DEplot(diff(y(x),x)=3*y(x)+1,[x,y],0..0.5,{[0,.1]},stepsize=.1,method=`Euler`,thickness=1):
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# 
> p2:=DEplot(diff(y(x),x)=3*y(x)+1,[x,y],0..0.5,{[0,.1]},stepsize=.05,method=`Euler`,thickness=2):
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> p3:=DEplot(diff(y(x),x)=3*y(x)+1,[x,y],0..0.5,{[0,.1]},stepsize=.01,method=`Euler`,thickness=3):
# 
> diffeq1:=diff(y(x),x)=3*y(x)+1; inits:=y(0)=.1;

                                     d
                        diffeq1 := ---- y(x) = 3 y(x) + 1
                                    dx

                               inits := y(0) = .1
> sol1:=dsolve({diffeq1,inits},y(x));

                   sol1 := y(x) = - 1/3 + .4333333333 exp(3 x)
> p4:=plot(rhs(sol1),x=0..0.4):
> 
> display([p1,p2,p3,p4]);
# HERE IS THE SAME PROBLEM SOLVED WITH IMPROVED EULER:
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# 
> p5:=DEplot(diff(y(x),x)=3*y(x)+1,[x,y],0..0.5,{[0,.1]},stepsize=.1,method=`impeuler`,thickness=1):
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# 
> p6:=DEplot(diff(y(x),x)=3*y(x)+1,[x,y],0..0.5,{[0,.1]},stepsize=.05,method=`impeuler`,thickness=2):
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> p7:=DEplot(diff(y(x),x)=3*y(x)+1,[x,y],0..0.5,{[0,.1]},stepsize=.01,method=`impeuler`,thickness=3):
> display([p4,p5,p6,p7]);
> ?DEtools[options];
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>